We prove that the topological entropy of the geodesic flow for a compact Riemannian manifold (M, g) decreases as the metric g evolves under the normalised Ricci flow provided that M admits a metric of constant negative sectional curvature, and g is in a sufficiently small C^2 neighbourhood of the constant curvature metric. More generally, the same phenomenon occurs if g satisfies a certain negative curvature pinching condition, where the pinching constant depends on both the dimension and the diameter of (M, g). This provides an affirmative answer to an open question posed in Manning's 2004 paper 'The volume entropy of a surface decreases along the Ricci flow'.